Competition over mates takes many forms and has far-reaching consequences for many organisms. Recent work suggests that relative reproductive rates of males and females, sperm competition and quality variation among mates affect the strength of sexual selection. Song, other display, body size, visual ornaments and material resource offerings are often sexually selected. There is much empirical evidence of mate choice, and its evolution is clarified by mathematical models. Recent advances in theory also consider costs of choice, effects of deleterious mutations, fast and slow evolution of preferences and preferred traits, and simultaneous preferences for several traits. Contests over mates are important; so is sperm competition, scrambles, endurance rivalry, and coercion. The latter mechanisms have received less attention than mate choice. Sexual selection may explain puzzling aspects of plant pollination biology.

The clinical success of the ABL tyrosine kinase inhibitor imatinib in chronic myeloid leukaemia (CML) serves as a model for molecularly targeted therapy of cancer, but at least two critical questions remain. Can imatinib eradicate leukaemic stem cells? What are the dynamics of relapse due to imatinib resistance, which is caused by mutations in the ABL kinase domain? The precise understanding of how imatinib exerts its therapeutic effect in CML and the ability to measure disease burden by quantitative polymerase chain reaction provide an opportunity to develop a mathematical approach. We find that a four-compartment model, based on the known biology of haematopoietic differentiation, can explain the kinetics of the molecular response to imatinib in a 169-patient data set. Successful therapy leads to a biphasic exponential decline of leukaemic cells. The first slope of 0.05 per day represents the turnover rate of differentiated leukaemic cells, while the second slope of 0.008 per day represents the turnover rate of leukaemic progenitors. The model suggests that imatinib is a potent inhibitor of the production of differentiated leukaemic cells, but does not deplete leukaemic stem cells. We calculate the probability of developing imatinib resistance mutations and estimate the time until detection of resistance. Our model provides the first quantitative insights into the in vivo kinetics of a human cancer.

We use a general additive quantitative genetic model to study the evolution of costly female mate choice by the "handicap" principle. Two necessary conditions must be satisfied for costly preference to evolve. The conditions are (i) biased mutation pressure on viability and (ii) a direct relationship between the degree of expression of the male mating character and viability. These two conditions explain the success and failure of previous models of the "handicap" principle. Our model also applies to other sources of fitness variation like migration and host-parasite coevolution, which cause effects equivalent to biased mutation.

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When the traditional assumption that the incidence rate is proportional to the product of the numbers of infectives and susceptibles is dropped, the SIRS model can exhibit qualitatively different dynamical behaviors, including Hopf bifurcations, saddle-node bifurcations, and homoclinic loop bifurcations. These may be important epidemiologically in that they demonstrate the possibility of infection outbreak and collapse, or autonomous periodic coexistence of disease and host. The possible mechanisms leading to nonlinear incidence rates are discussed. Finally, a modified general criterion for supercritical or subcritical Hopf bifurcation of 2-dimensional systems is presented.

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We introduce a demographic model for a local population of sessile marine invertebrates that have a pelagic larval phase. The processes in the model are the settling of larvae onto empty space, and the growth and mortality of the settled organisms. The rate of settlement per unit of unoccupied space is assumed to be determined by factors outside of the local system. The model predicts the number of animals of each age in the local system through time. The model is offered in both discrete and continuous—time versions. The principal result is that the growth of the settled organisms is destabilizing. In the model, there is always a state where recruitment balances mortality. However, growth can interfere with recruitment and can destabilize this steady state, provided also that the settlement rate is sufficiently high. The model suggests that two qualitatively distinct pictures of population structure result, depending on the settlement rate. In the high settlement limit, the intertidal landscape is a mosaic of cohorts, punctuated with occasional gaps of vacant substrate. In the low settlement limit, the intertidal landscape has vacant space and organisms of all ages mixed together and spatial variation in abundance is caused by microgeographic variation in settlement and mortality rates.

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